Almost any distro will work, but you'll probably have to use a lightweight window manager. Each installer is a little different as to how you select which packages to install. But basically you want to disable Gnome/KDE at install and use XFCE, windowmaker, icewm, etc. instead. If you want to try Ubuntu, download and use Xubuntu (uses XFCE instead of Gnome/KDE). Or buy more RAM.
Now if you don't need a gui, I'd say install Debian
from a netinstall CD. You'll get a text only prompt. Here's a good site
on how to setup a lot of things from the command line. Then if you want a desktop, you can always aptitude install x-window-system icewm synaptic
as root and get yourself a pretty gui.
You definitely don't need a FAT32 partition for network sharing, only if Windows and Linux on are on the same machine. Over a network, all the files get converted to bits and bytes for sending and decoded by the OS. You just need to setup Samba.